Axiom ax-precex 7730

Description: Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7688. (Contributed by Jim Kingdon, 6-Feb-2020.)

Assertion

Ref	Expression
ax-precex	⊢ ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))

Distinct variable group:   𝑥,𝐴

Detailed syntax breakdown of Axiom ax-precex

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 7619	. . . 4 class ℝ
3	1, 2	wcel 1480	. . 3 wff 𝐴 ∈ ℝ
4		cc0 7620	. . . 4 class 0
5		cltrr 7624	. . . 4 class <ℝ
6	4, 1, 5	wbr 3929	. . 3 wff 0 <ℝ 𝐴
7	3, 6	wa 103	. 2 wff (𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴)
8		vx	. . . . . 6 setvar 𝑥
9	8	cv 1330	. . . . 5 class 𝑥
10	4, 9, 5	wbr 3929	. . . 4 wff 0 <ℝ 𝑥
11		cmul 7625	. . . . . 6 class ·
12	1, 9, 11	co 5774	. . . . 5 class (𝐴 · 𝑥)
13		c1 7621	. . . . 5 class 1
14	12, 13	wceq 1331	. . . 4 wff (𝐴 · 𝑥) = 1
15	10, 14	wa 103	. . 3 wff (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)
16	15, 8, 2	wrex 2417	. 2 wff ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1)
17	7, 16	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 0 <ℝ 𝐴) → ∃𝑥 ∈ ℝ (0 <ℝ 𝑥 ∧ (𝐴 · 𝑥) = 1))

This axiom is referenced by:  recexre  8340  recexgt0  8342